On an irreducibility type condition for the ergodicity of nonconservative semigroups

Bertrand Cloez,Pierre Gabriel

doi:10.5802/crmath.92

Abstract

We propose a simple criterion, inspired from the irreducible aperiodic Markov chains, to derive the exponential convergence of general positive semi-groups. When not checkable on the whole state space, it can be combined to the use of Lyapunov functions. It differs from the usual generalization of irreducibility and is based on the accessibility of the trajectories of the underlying dynamics. It allows to obtain new existence results of principal eigenelements, and their exponential attractiveness, for a nonlocal selection-mutation population dynamics model defined in a space-time varying environment.

Highlights

We are interested in the long time behavior of positive semigroups in weighted signed measures spaces
More precisely for a weight function V : X → (0, ∞) we denote by M+(V ) the set of positive measures on X which integrate V and we define the space of weighted signed measures as
We present sufficient irreducibility type conditions which, possibly combined to Lyapunov type conditions, ensure the so-called asynchronous exponential behavior of the semigroup: μMt ∼ eλt μ(h)γ as t → ∞, where λ ∈ R, h is a positive function, and γ is a positive measure

Summary

The main result

We are interested in the long time behavior of positive semigroups in weighted signed measures spaces. There are few examples of Markov processes on infinite state spaces which are irreducible since in most cases the probability of reaching a point y at a deterministic time starting from an arbitrary position x is zero. Condition (H2) is an aperiodicity assumption, reminiscent from the aperiodicity of Markov chains It can be seen as a coupling condition since it means that, uniformly in x and x in X , the trajectories issued from x and x intersect at time τ with positive probability (the intersection point being on a trajectory issued from some y ∈ X in the time interval [0, τ]). (3) is similar to [2, Assumption (H1)] with the difference that the coupling measure ν is independent of x, x It is not an issue since the proof in [2] works with a family (νx,x ).

An application in population dynamics

Conclusion

Methods